Neural Operator Variational Inference based on Regularized Stein Discrepancy for Deep Gaussian Processes

Deep Gaussian Process (DGP) models offer a powerful nonparametric approach for Bayesian inference, but exact inference is typically intractable, motivating the use of various approximations. However, existing approaches, such as mean-field Gaussian assumptions, limit the expressiveness and efficacy of DGP models, while stochastic approximation can be computationally expensive. To tackle these challenges, we introduce Neural Operator Variational Inference (NOVI) for Deep Gaussian Processes. NOVI uses a neural generator to obtain a sampler and minimizes the Regularized Stein Discrepancy in L2 space between the generated distribution and true posterior. We solve the minimax problem using Monte Carlo estimation and subsampling stochastic optimization techniques. We demonstrate that the bias introduced by our method can be controlled by multiplying the Fisher divergence with a constant, which leads to robust error control and ensures the stability and precision of the algorithm. Our experiments on datasets ranging from hundreds to tens of thousands demonstrate the effectiveness and the faster convergence rate of the proposed method. We achieve a classification accuracy of 93.56 on the CIFAR10 dataset, outperforming SOTA Gaussian process methods. Furthermore, our method guarantees theoretically controlled prediction error for DGP models and demonstrates remarkable performance on various datasets. We are optimistic that NOVI has the potential to enhance the performance of deep Bayesian nonparametric models and could have significant implications for various practical applications

Double Normalizing Flows: Flexible Bayesian Gaussian Process ODEs Learning

Recently, Gaussian processes have been utilized to model the vector field of continuous dynamical systems. Bayesian inference for such models \cite{hegde2022variational} has been extensively studied and has been applied in tasks such as time series prediction, providing uncertain estimates. However, previous Gaussian Process Ordinary Differential Equation (ODE) models may underperform on datasets with non-Gaussian process priors, as their constrained priors and mean-field posteriors may lack flexibility. To address this limitation, we incorporate normalizing flows to reparameterize the vector field of ODEs, resulting in a more flexible and expressive prior distribution. Additionally, due to the analytically tractable probability density functions of normalizing flows, we apply them to the posterior inference of GP ODEs, generating a non-Gaussian posterior. Through these dual applications of normalizing flows, our model improves accuracy and uncertainty estimates for Bayesian Gaussian Process ODEs. The effectiveness of our approach is demonstrated on simulated dynamical systems and real-world human motion data, including tasks such as time series prediction and missing data recovery. Experimental results indicate that our proposed method effectively captures model uncertainty while improving accuracy.